Singular Manual: purityFiltration

7.5.23.0. purityFiltration

Procedure from library purityfiltration.lib (see purityfiltration_lib).

Usage:: purityFiltration(S), S matrix with entries of an Auslander regular ring D
Return:: a list T of two lists, purity filtration of the module M=D^q/D^p(S^t)
Purpose:: the first list T[1] gives a filtration {M_i} of M,
where the i-th entry of T[1] gives the representation matrix of M_(i-1).
the second list T[2] gives representations of the factor Modules,
i.e. T[2][i] gives the repr. matrix for M_(i-1)/M_i

Example:

LIB "purityfiltration.lib";
ring D = 0,(x1,x2,d1,d2),dp;
def S=Weyl();
setring S;
int i;
matrix R[3][3]=0,d2-d1,d2-d1,d2,-d1,-d1-d2,d1,-d1,-2*d1;
print(R);
==> 0, -d1+d2,-d1+d2,
==> d2,-d1,   -d1-d2,
==> d1,-d1,   -2*d1  
list T=purityFiltration(transpose(R));
// the purity filtration of coker(M)
print(T[1][1]);
==> 0, -d1+d2,-d1+d2,
==> d2,-d1,   -d1-d2,
==> d1,-d1,   -2*d1  
print(T[1][2]);
==> d2,   d2,
==> d1-d2,0, 
==> d2,   d1 
print(T[1][3]);
==> 1,0, 
==> 0,d2,
==> 0,d1 
// factor modules of the filtration
print(T[2][1]);
==> 0, 1,1,
==> -1,0,1 
print(T[2][2]);
==> 1,    1,
==> d1-d2,0 
print(T[2][3]);
==> 1,0, 
==> 0,d2,
==> 0,d1

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